Section 18.1-2. in the Next 2-3 Lectures We Will Have a Lightning Introduction to Representations of ﬁnite Groups

Home , Invariant subspace

Section 18.1-2. In the next 2-3 lectures we will have a lightning introduction to representations of ﬁnite groups. For any vector space V over a ﬁeld F , denote by L(V ) the algebra of all linear operators on V , and by GL(V ) the group of invertible linear operators. Note that if dim(V ) = n, then L(V ) = Matn(F ) is isomorphic to the algebra of n×n matrices over F , and GL(V ) = GLn(F ) to its multiplicative group.

Deﬁnition 1. A linear representation of a set X in a vector space V is a map φ: X → L(V ), where L(V ) is the set of linear operators on V , the space V is called the space of the representation. We will often denote the representation of X by (V, φ) or simply by V if the homomorphism φ is clear from the context. If X has any additional structures, we require that the map φ is a homomorphism. For example, a linear representation of a group G is a homomorphism φ: G → GL(V ).

Deﬁnition 2. A morphism of representations φ: X → L(V ) and ψ : X → L(U) is a linear map T : V → U, such that ψ(x) ◦ T = T ◦ φ(x) for all x ∈ X. In other words, T makes the following diagram commutative

φ(x) V / V

T T ψ(x) U / U An invertible morphism of two representation is called an isomorphism, and two representations are called isomorphic (or equivalent) if there exists an isomorphism between them.

Example. (1) A representation of a one-element set in a vector space V is simply a linear operator on V . Two such representations are isomorphic if two operators have the same Jordan Normal Form. (2) For any vector space, identity maps id: GL(V ) → GL(V ) and id: L(V ) → L(V ) give tautological representations of the group GL(V ) and the algebra L(V ) respectively. (3) For each representation φ: X → L(V ) one can deﬁne its dual representation

φ∗ : X → L(V ∗), φ∗(x)(α)(v) = α(φ(x)(v))

for all x ∈ X, v ∈ V , and α ∈ V ∗. (4) The trivial representation of a group G is a homomorphism φ: G → GL(V ), where V is a 1-dimensional vector space over the ﬁeld F , and φ(g) = 1 ∈ GL(V ) = F × for all g ∈ G. n 2 −1 (5) Recall the dihedral group Dn = r, s | r = s = 1, rs = sr . The following map

cos (2π/n) − sin (2π/n) 0 1 r 7→ , s 7→ sin (2π/n) cos (2π/n) 1 0

deﬁnes a 2-dimensional representation φ: Dn → GL(2, R). 1 2

(6) The following map gives a representation of a symmetric group S4: 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 (12) 7→   , (23) 7→   , (34) 7→   . 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0

(7) There is a natural representations of Sn on the space of polynomials in n variables:

(σf)(x1, . . . , xn) = f(xσ(1), . . . , xσ(n)). Note, that symmetric polynomials stay invariant under this action. (8) Any Galois group Gal(K/F ) comes with its representation in K over the ﬁeld F . Deﬁnition 3. An invariant subspace of a representation φ: X → L(V ) is a subspace W ⊂ V invariant under φ(x) for all x ∈ X. An invariant subspace gives rise to a subrepresentation

φW : X → L(W ), φW (x) = φ(x)|W , and a factor representation

φV/W : X → L(V/W ), φV/W (x)(v + W ) = φ(x)(v). Remark 4. If one chooses a basis of the space V of a representation φ: X → L(V ) in such a way that the ﬁrst k vectors span an invariant subspace W ⊂ V , then any operator φ(x) is written in matrix form as φ (x) ∗ φ(x) = W 0 φV/W (x) Example. n (1) Let Sn act on F by permuting basis vectors, as in example (4) above. Then the subspace {a, a, . . . , a) | a ∈ F } ⊂ F n is a 1-dimensional invariant subspace. There is a complementary invariant subspace of dimension n − 1 deﬁned by

{(a1, . . . , an) | a1 + ··· + an = 0} . Note that F n is isomorphic to the direct sum of these two subspaces. d (2) For any positive integer d, the subspace Poln of polynomials in n variables of degree d in the space Poln of all polynomials of n variables gives a subrepresentation of Sn. Deﬁnition 5. (1) A representation is irreducible (or simple) if it (or rather the underlying vector space) has no invariant subspaces other than 0 and itself, otherwise it is reducible. (2) A representation is indecomposable if it can not be written as a direct sum of two invariant subrepresentations. (3) A representation is completely reducible if it can be written as a direct sum of irreducible representations. Example. The representation of a 1-element set given by a 2-dimensional space V = F 2 and an operator a b A = 0 c is reducible, since the subspace he1i ⊂ V is invariant, however it is indecomposable unless b = 0. If b = 0, the representation V is completly reducible: V = he1i ⊕ he2i. Exercise 6. If T : V → U is a morphism of representations of X, then ker(T ) ⊂ V and im(T ) ⊂ U are invariant subspaces. 3

Corollary 7. If V and U are irreducible representations of X, any morphism T : V → U is either 0 or an isomorphism. Proof. Since V is irreducible, the subrepresentation ker(T ) is either 0, in which case T is injective, or coincides with the whole V , in which case T = 0. Since U is irreducible, we must now have im(T ) = U unless T = 0, which forces T to be an isomorphism. Corollary 8 (Schur’s lemma). Any endomorphism T : V → V of an irreducible representation V of X over an algebraically closed ﬁeld is a scalar, that is T = λ · Id for some λ ∈ F .

Proof. Let T ∈ EndX (V ) = HomX (V,V ) be an endomorphism of V . Then for any λ ∈ F , so is T −λ·Id. Choosing λ to be an eigenvalue of T (which we can always do over an algebraically closed field) we see that ker(T − λ · Id) is a subrepresentation of X in V . Therefore, since V is irreducible, we must have T = λ · Id. Corollary 9. Let φ: X → L(V ) and ψ : X → L(U) be a pair of irreducible representations over an algebraically closed field. Then any morphisms T,S : V → U differ by a scalar factor. Proof. If one of the two morphisms is zero, the statement is obvious. Otherwise, both morphisms are isomorphisms, hence ST −1 is an endomorphism of an irreducible representation −1 U which implies ST = λ Id. Corollary 10. Any irreducible representation of an abelian group over an algebraically closed field is 1-dimensional. Proof. If G is abelian and φ: G → GL(V ) is a representation of G, operators φ(g) commute for all g ∈ G, hence any of them can be viewed as an endomorphism of V . If V is irreducible, By Schur’s lemma, all of them must be scalar, therefore any subspace is invariant and representation V can be irreducible if and only if dim(V ) = 1. 2 Example. The 2-dimensional plane R is an irreducible representation of the abelian group SO(2, R) of plane rotations cos θ − sin θ G = θ ∈ R . sin θ cos θ However, over the field of complex numbers, we see that cos θ − sin θ eiθ 0 ∼ . sin θ cos θ 0 e−iθ Proposition 11. Every subrepresentation and factor representation of a completely reducible representation is completely reducible. Proof. Let V be a completely reducible representation, and U be its subrepresentation. 0 0 Then for any U1 ⊂ U, there exists an invarinat subspace V ⊂ V such that V ' U1 ⊕ V . 0 Setting U2 = V ∩ U we see that U2 is invariant and U ' U1 ⊕ U2. Now, let π : V → V/W be a canonical projection to a factor representation, and U1 ⊂ V/W −1 be the subrepresentation. Then V1 = π (U1) is an invariant subspace of V , which con- tains W . Now, we have V = V1 ⊕ V2 for some invariant subspace V2, which implies that V/W ' U1 ⊕ U2 where U2 = π(V2) is an invariant subspace in V/W . Proposition 12. If a representation φ: X → L(V ) is completely reducible, then V can be written as a direct sum of minimal (nonzero) irreducible subspaces. Conversely, if V can be written as a sum of minimal invariant subspaces V = V1 + ··· + Vn, then V is completely reducible. 4

Proof. The ﬁrst statement is immediate: choose any minimal invariant subspace V1 ⊂ V and write V = V1 ⊕V2, continue the same process for V2. Now, let V = V1 +···+Vn for some minimal invariant subspaces Vi ⊂ V , and U ⊂ V be any invariant subspace. Then U ∩ Vj is either 0 or coincides with Vj for any j = 1, . . . , n. Setting 0 X U = Vj j:U∩Vj =∅ 0 we get V = U ⊕ U . Deﬁnition 13. A sum of representations φ: X → L(V ) and ψ : X → L(U) is a representation φ + ψ : X → V ⊕ U, (φ + ψ)(x) = (φ(x), ψ(x)). Remark 14. In matrix form we have φ(x) 0 (φ + ψ)(x) = 0 ψ(x) Corollary 15. (1) A representation is completely reducible if and only if it is isomorphic to a sum of irreducible subrepresentations:

V = V1 ⊕ · · · ⊕ Vn, φ = (φ1, . . . , φn). (2) In that case, any subrepresentation and factor representation of V is isomorphic to a sum of some of the representations Vj.

Proof. Left as an exercise.

Deﬁnition 16. Note that in the above corollary, some of the irreducible summands Vj might be isomorphic. (1) Let S be an irreducible subrepresentation of a completely reducible representation V , then the S-isotypic component of V is the subspace VS ⊂ V deﬁned as M VS = Vj.

j | Vj 'S (2) A representation V is isotypic if it has single isotypic component. (3) We will say that the representation V has simple spectrum if it can be written as a direct sum of pairwise non-isomorphic irreducible representations, equivalently if all of its isotypic components are irreducible. Corollary 17. Let φ: X → L(V ) be a completely reducible representation with simple spectrum, and V1,...,Vn be its irreducible subrepresentations. Then the following holds. (1) The decomposition of V into a sum of irreducible subrepresentations Vj is unique. (2) If the base ﬁeld F is algebraically closed, then every endomorphism of V takes the form T (x) = λjx, for any x ∈ Vj.

Proof. Left as an exercise. It is convenient to describe isotypic representations in the following way. Let φ: X → L(V ) be a representation, and Z be any vector space (over the same ﬁeld). Deﬁne a representation φ˜: L(V ⊗ Z), φ˜(x)(v ⊗ z) = φ(x)(v) ⊗ z. 5

If {z1, . . . zn} is a basis of Z we get

V ⊗ Z ' (V ⊗ z1) ⊕ · · · ⊕ (V ⊗ zn), which provides a decomposition of an isotypic representation V ⊗ Z into irreducible subrepresentations. Proposition 18. Let φ: X → L(U) be an irreducible representation of X over an algebraically closed ﬁeld F , Z be a vector space over F , and ψ : X → L(U ⊗ Z) be the isotypic representation of X discussed above. Then every X-invariant subspace of U ⊗ Z is of the form U ⊗ Z0, for some subspace Z0 ⊂ Z. Proof. It is enough to prove the statement for a minimal invariant subspace W ⊂ U ⊗ Z. Any element w ∈ W can be written as

w = S1(w) ⊗ z1 + ··· + Sn(w) ⊗ zn, where Sj : W → U are certain morphisms between representations (W, ψ|W ) and (U, φ). We know that any such isomorphisms have to diﬀer by scalar multiples, hence Sj = λjS for some morphism S : W → U and λ1, . . . , λn ∈ F . This way we obtain

w = S(w) ⊗ (λ1z1 + ··· + λnzn), and therefore W = U ⊗ (λ1z1 + ··· + λnzn). Theorem 19. Let φ: X → L(V ) be an irreducible representation of X over an algebraically closed ﬁeld F . Then the subalgebra of L(V ) generated by operators φ(x), x ∈ X coincides with L(V ) unless dim(V ) = 1 and φ = 0. Proof. Recall the following isomorphism of vector spaces: ∗ V ⊗ V ' L(V ), v ⊗ α 7−→ lα,v ∗ where `α,v(u) = α(u)v for all α ∈ V and u, v ∈ V . Note that under this isomorphism, for any operator T ∈ L(V ) we have

T `α,v = `α,T (v) = `T ∗(α),v, (∗) where T ∗ ∈ L(V ∗) deﬁned by T ∗(α)(v) = α(T v). Due to a canonical bijection between subspaces of V and V ∗, which sends each subspace U ⊂ V to its annihilator Ann(U) ⊂ V ∗, we see that the dual representation φ∗ : X → L(V ∗) is also irreducible. Let us now deﬁne representations Tl and Tr of X on the space L(V ), precisely Tl(x)(A) = φ(x)A and Tr(x)A = Aφ(x).

Thanks to the formula (∗), representations Tl and Tr are isotypic (we can think that they act on only one tensor factor in V ⊗ V ∗). Note now that the subspace φ(X) ⊂ L(V ) is invariant under both Tl and Tr. Using the previous Proposition, we conclude that φ(X) can be written simultaneously as V ⊗ W0 and ∗ ∗ as V0 ⊗ V , where V0 and W0 are subspaces of V and V respectively. The latter can only happen if φ(X) = L(V ) or φ(X) = 0 in which case one has to assume dim(V ) = 1 since otherwise V would not be irreducible. 6

Theorem 20. Any representation of a finite group G over a field F such that char(F ) does not divide |G| is completely reducible. Proof. Let G be a finite group, and φ: G → GL(V ) be its representation over a field F with char(F ) not dividing |G|. It is enough to show that for any invariant subspace U ⊂ V there exists a complementary invariant subspace W ⊂ V such that V ' U ⊕ W . The latter is equivalent to finding a projector π from V onto U, which is a morphism of representations of G. Indeed, recall that a projector is a linear operator P : V → U such that P (V ) ⊂ U and P (u) = u for any u ∈ U. Then setting W = ker(P ) we obtain an invariant subspace W ⊂ V satisfying V ' U ⊕ W. Note that the space of all projectors P : V → U forms a linear subspace S in the space L(V ) of all linear operators on V . Consider an action of G on L(V ) by conjugation, that is g ◦ A = φ(g)Aφ(g)−1 for all g ∈ G and A ∈ L(V ). Since, U is a subrepresentation of G we see that the subspace S ⊂ L(V ) is preserved under this action. Let us choose any projector P0 ∈ S and define P to be the “centre of mass” of the orbit of P0 under the action of G: 1 X P = (g ◦ P ). |G| 0 g∈G Then the projector P ∈ S is invariant under the adjoint action of G, or equivalently, commutes with all operators φ(g) ∈ L(V ). Finally, let us discuss representations of the group algebra FG of a finite group G, where F is a field. Recall that an algebra A over a ring R is an R-module, which is a ring itself, in particular if R = F is a field, then an algebra A over F is a vector space over F , with a ring structure. As a vector space, algebra FG consists of linear combinations X agg where ag ∈ F, g∈G addition on FG is defined component-wise, and multiplication has the form

agg · ahh = agah(gh). In other words, FG is an algebra over the field F with basis vectors labelled by elements of the finite group G, and with multiplication defined from that of G. As we discussed in the beginning of this lecture, a representation of an algebra A (over the ring R) is a homomorphism φ: A → L(V ) which respects the operations on A. In other words, a representation of A is simply an A-module V (which then forced to be an R-module itself). Now, it is easy to see that there is a bijection between representations of the group G and its group algebra FG, moreover this bijection respects subrepresentations, factor representations, etc. Indeed, given a representation φ: G → GL(V ) of the group G, we define (and denote by the same symbol) a representation X X φ: FG → L(V ), φ( agg) = agφ(g). g∈G g∈G Conversely, every representation of FG yields a representation of G when restricted to the basis vectors of FG. 7

Example. (1) The trivial representation of G corresponds to a 1-dimensional representation of FG where

(agg)(v) = agv for any ag ∈ F, g ∈ G, v ∈ V. (2) Considering FG as a (left) module over itself, we obtain a (left) regular representation of G, which is a representation of dimension |G| defined by X X g( ahh) = ah(gh) for all ah ∈ F, g, h ∈ G. h∈G h∈G The bijection described above allows us to study representations of finite groups by study- ing those of finite-dimensional associative algebras.